"would become" is more commonly used when describing the effect of a single conditional:
If I could get my hands on that weapon, then I would become the most powerful person in the world!
To phrase your context in this manner:
If they attended that school, then they would become great musicians.
You can use "would become" in the first part of the conditional:
If you would become the most powerful person in the world, then I must kill you.
But your example is trying to use it in a nested hypothetical:
What if you would become the most powerful person in the world?
Using "would become" here assumes a second "if":
What if you would become the most powerful person in the world if you got your hands on that weapon?
Your example's full context provides the second "if" in the first sentence. Merging the sentences together is stylistically ugly but it can help highlight what is happening:
What if they would become great musicians [if they attended that school]?
Note that the "if" doesn't have to be a literal "if":
What if they would become great musicians [as a result of attending that school]?
What if they would become great musicians [because they attended that school]?
Therefore, your first example is correct usage.
Your second example uses "became" which does not automatically assume a second conditional. You could simply ask:
What if they became great musicians?
The presence of the first sentence in your context does imply such a second conditional and this is perfectly acceptable:
Do you think they shouldn't attend that school? What if they became great musicians?
It is clear from the context that the "became" clause is implying that they would become musicians as a result of attending the school. Technically this association is unnecessary and you could theoretically read the question as:
Do you think they shouldn't attend that school? I want them to become lawyers by attending that school. What if they became great musicians [because they didn't attend that school]?
But this is less common and is more likely to confuse readers. So, in the end, the second example is also correct usage and conveys the same meaning as the first example.