What do you mean by your statement that "believing in Christ is 'the work of God'"?
Coming to Christ equals believing in Christ in this context (v. 35). Coming is attributed as consequential to first being given by the Father which is a work of God (Jn. 6:29-39). Coming is attributed as consequential to being drawn by the Father which is a work of God. The lack of being drawn is attributed to remaining in the state of unbelief (Jn. 6:64-65). At minimum, the work of God in giving and drawing is explicitly provided as prerequisites for coming to Christ by faith and both are the work of God. Therefore, the following context from John 6:29 does in fact teach that coming to faith in Christ is the work of God = "This is the work of God that ye believe."
That the act of believing itself is caused by God?
As far as ultimate cause? Yes! That is the good work he has begun in us (Philip 1:6) as to believe in Christ is given to us (Philip. 1:29) working in us "both to will and to do" of His good pleasure (Philip. 2:13). Not that we are
forced to believe, but that drawing by God is the internal work of God (v. 45) that transforms the inward natural disposition from hating to loving light, from a mental state of unbelief to belief and therefore both mechanisms that control the will are changed so that the state of the will which is at war against God is changed to willing submission to God or the complete internal transformation of the heart through the new birth.
You use quotes. Does that mean you're quoting 6:29 to substantiate your claim? If so, that's certainly not what 6:29 means.
In verse 27 the labor Jesus refers to was their following him across the sea in order to obtain more physical food. He tells them that their motivation and energy in seeking him out was misdirected. They ought to have expended that energy in seeking him for food that wrought eternal life, that Christ would
GIVE to them. He is not implying they have the inherent ability to achieve that task but only that would be the right motive and expenditure of energy. Indeed, Jesus says that eternal life is something he will "GIVE" rather than something achieved by their works. However, in response to something that Jesus said was to be GIVEN to them, they claim they can "do" whatever is necessary to obtain eternal life if they are but told what to "do" and they will "work the works of God. This is the precise point where expositors of this passage mess up. They fail to see to see that the audience is responding by claiming inherent ability to "do" the "works of God" to obtain eternal life when Jesus had just said it was something that was GIVEN. Hence, they fail to see the clash between Christ's assertion that eternal life is a gift versus the assertion by his audience that they are able to obtain it by works. Verse 29 is Christ's response to this clash over gift versus works. Coming to Christ by faith to obtain eternal life is not by working the works (plural) of God but "this is the work (singular) of God that ye believe." The work of God (vv. 37-45) was not performed in regard to them and so they do not believe (v. 36).
They wrongly interpret the words of Christ in verse 27 to mean that he is telling them they
can obtain eternal life by "labor". He is no more telling them they are able to do that to obtain eternal life than he was telling the Rich Young Ruler or the Pharisee that they are
able to keep the Law to obtain eternal life. Keeping the law would obtain eternal life IF they were able but they are not. However, directing them to keep the law for eternal life is the right thing to do in regard to their question "what must I DO" to obtain eternal life. Likewise, telling them to labor for eternal life is the right thing to do based upon the heart attitude which is later expressed in explicit language of works. That would be the right direction of their labors if their labors could obtain eternal life but they lack the ability (Jn. 6:44). Eternal life cannot be obtained by works but is a GIFT that the Son GIVES to all coming to him by faith which is consequential of God's work in giving and drawing. He is merely rebuking them for expenditure of motive and labor toward the wrong thing. However, by directing their expenditure of motive toward the right thing, he is not claiming they are ABLE to obtain it by their labor but only that is the proper direction of motive and labor. Later, he will deny their ABILITY to obtain it (v. 44) as that ability is the product of the "work of God" in drawing.
Most interpreters fail to see that Christ is actually denying they have inherent ability to "do" the "works of God" for eternal life. Instead, he is asserting that what must be done to obtain eternal life is only something God can do - "This is the work of God that ye believe." Notice they ask what "works" plural they can "do" but Christ responds with "work" singular. In verse 30 they continue to make their claim that they have inherent ability to believe "that we MAY believe" (although Christ will later say "no man can come" apart from the work of God, which is the assertion of universal inability to believe).
In verses 31-33 Christ makes it clear that he is the manna sent by the Father from heaven to give eternal life. In verse 34 they ask him to give them this manna to live forever. Jesus tells them how to receive it in verse 35 but then, before they can reply, tells them they are in a state of unbelief in verse 36 even though they had seen him perform the very kind of miracle they asked for in verse 30 in the feeding of the 5000. Thus the problem of unbelief is not in seeing the miraculous as they claimed in verse 30 but in the fact that "the work of God" had not been performed in regard to them (vv. 37-39) so that they could come to Christ. Now, look at the contrast between verse 36 and verse 40. The former have seen (horao) or perceived him from an external point of view whereas the latter see him (theoreo) from an inward experiential point of view because they were "given" by the Father to come to Christ. This experiential point of view is later defined in the meaning of drawing in verse 45 or the internal instruction by God so that the person experiences hearing and learning from God.
You may choose to reject my interpretation of John 6:29 but you cannot possibly deny that coming to Christ by faith is
the work of God described in John 6:37-45 where giving and drawing are necessary antecedents for coming to Christ by faith and thus "
this is the work of God that ye believe." The reason for the unbelief of his audience in verse 36 is the lack of the Father's work in giving them (vv. 37-40) and drawing them (vv. 44-45) as that is the explicit reason stated for the unbelief of those in verses 63-66.