Is belief a work into salvation? Is the placing of one's faith in Christ for salvation, a work?  - Can I produce faith on my own?
How does the Bible answer that question?
You have asked for a biblical answer, and there are certainly different positions on this.  Some who believe that works are essential for salvation will refer to belief/faith as a work.  Indeed, if faith can be classified as a work, then requiring additional works for salvation seems more reasonable.  Yet, there appears to be good reason to distinguish faith from works.
Ephesians 2:8-9 is, of course, a key passage:

For by grace you have been saved through faith. And this is not
your own doing; it is the gift of God, 9 not a result of works,
so that no one may boast.  Ephesians 2:8-9 ESV

Salvation comes through faith, but salvation is not our own doing, but a gift of God.  Salvation is not a result of works (but of faith).  Faith, then, is distinguished from works.
(Some assert that faith is the gift of God, &quot;this&quot; does not match &quot;faith&quot; in gender, so that doesn't really fit.  It would be like saying, &quot;my brother and his wife came over, and he got a new job--you would know I am speaking about my brother getting a new job because of the gender agreement)
There are quite a few other passages that demonstrate salvation is by faith, but not by works:

But now the righteousness of God has been manifested apart from the
law, although the Law and the Prophets bear witness to it— 22 the
righteousness of God through faith in Jesus Christ for all who
believe. For there is no distinction: 23 for all have sinned and
fall short of the glory of God... Romans 3:21-23 ESV
For we hold that one is justified by faith apart from works of the law.  Romans 3:28 ESV
16 “For God so loved the world, that he gave his only Son, that
whoever believes in him should not perish but have eternal life... 18
Whoever believes in him is not condemned, but whoever does not believe
is condemned already, because he has not believed in the name of the
only Son of God.  John 3:16, 18 ESV
We ourselves are Jews by birth and not Gentile sinners; 16 yet we know
that a person is not justified by works of the law but through
faith in Jesus Christ, so we also have believed in Christ Jesus, in
order to be justified by faith in Christ and not by works of the law,
because by works of the law no one will be justified.  Galatians 2:15-16 ESV

Addendum
I was reading in Romans last night and came across this verse that specifically distinguishes works from faith:

And to the one who does not work but believes in him who justifies the ungodly, his faith is counted as righteousness.  Romans 4:5 ESV

Paul speaks of the one who does not work, but believes--that person's faith is counted as righteousness.  Thus, faith is not a work, but is contrasted with works.