My teachers are 'Magisters.'
My king is his 'Majesty.'
My dragon is 'Majestic.'
Is there some etymological link between the Latin word for 'teacher' and the words we use for exaltation?
As an alternative perspective (which I hope will be corrected/supplemented by those who have more linguistic knowledge), I don't think you need to go quite so far back to see the relation of these words. Though the actual developmental relationship traces back to PIE, as Ethan shows, there's a simple connection that exists in Latin and which any reflective Roman would have grasped, namely:
Of course, magis and maior are not regularly formed, and hence we need to look further back to trace their common ancestry. But they both have a clear relationship to one of the most common adjectives in Latin: magnus.
If you limit your etymological range to Latin alone, no.
These three English words which you've listed coincidentally happen to have similar pronunciation in their first few syllables. Both magister and majesty find their roots in different Latin words:
majesty < mājestās
majestic < majesty- + -ic
magister < magister
The Latin words themselves, however, from which these English words originate, do share the same roots; both ultimately arrive from the same Proto-Indo-European adjective, *méǵh₂s:
mājestās < major + -tās < *magjōs < *méǵh₂yōs < *méǵh₂s + *-yōs
magister < magis + -tero < magnus < *magnos < *méǵh₂s