First of all, the methanol and acetyl chloride will make methyl acetate and HCl. Salicilic acid could then get esterified at the carboxylic acid site with the methanol through acid catalysis. The methylation of the phenol group could proceed through acid catalysis as well, but is less common.
The answer is NO, or Did you have find something else in lab! . However, you may end up with methyl o-acetylsalicylate [Asprin methyl ester].
On the other hand to have the transformation you are proposing, you should use dimethyl sulfate [or MeI or dimethyl carbonate], K2CO3 in acetone at 50C-reflux.