So would the warrant show up on the screeen of the drivers license office??

I will assume, that "wood" is would show up.
And warrant is warrant
And license is license

But for the life of me, I don't know what type of license is a
Alta license. ( I know it is not attorney)

But I will assume since you said drivers license it is for some country I don't know ( ALTA)
So I can't tell you how they do it there.
But it may show up yes, and at the least, since you have to prove and provide your address, they will know where to find you ( at least in most places in the US)