You can burn any kind of seeds,but most folks would rather work with wood chunks,that often will be more consistent.
Considering, you can buy enough wood chunks at the box stores,for $15 ,to last a couple years,most folks don't figure it is worth messing with,when you could be learning to cook the meats.
As far as maybe quick cooking fish/breast fillets,you won't use much mesquite anyway-you are just adding one more variable,when you could be learning to cook the product ,correctly.
Mesquite is pretty much used by grillers to get the high BTU s,for searing meats in fast cooking.
Now ,you need to adjust the loss in BTU s from burning wood,to trying to cook with a moist nut?
Most experienced cooks will advocate learning to cook the meat correctly,which is 97% of your success,and not get sidetracked with the 100 things that don't really count.
Just my $0.02