The sound /a/ in Mi’gmaq is pronounced low and in the back of the mouth.  When you add a /g/ after it, the /a/ sound pulls the /g/ down low and back into your throat, giving it that very distinctive guttural sound that we write with /q/.

(There are still some words—like jagej ‘lobster’—that do have /ag/, but as a general rule, /ag/ almost always comes out pronounced as /aq/, especially, as Erin notes, before a consonant or the end of the word.)

So for example, the common ending -g, which you may know from these two words

tap’tang    ‘potatoes’                                                tap’tan    ‘potato’

comes out as -q after the -a- that some words have before the ending:

ga’taq        ‘eels’                (comes from     ga’ta-g)            ga’t        ‘eel’
muinaq    ‘bears’                (comes from     muina-g)            muin        ‘bear’

As Erin notes, /o/ will do the same thing, so that /og/ comes out as /oq/.  This is probably because /o/ mostly comes from /a/-sounds that have fused together with a nearby /w/ or /gw/ sound.