Neither of these two words is a name of a god, which normally does not matter, because both the Christians and the Muslims believe that there exists only one god, so it is not necessary to have a proper name for that god.
Obviously, whenever a Christian believer or a Muslim believer speaks about the "God", they speak about the only god they believe in, so it makes no sense to ask them which god they mean.
A non-believer may speak about the God of the Muslims and about the God of the Christians, to make clear the intended meaning, and the latter phrase might be translated into Arabic with words meaning "the Allah of the Christians".