Core Rulebook > Equipment > Magic Items (pg. 224) wrote:
You don’t need to know the spell within a spell gem to use it, but the spell must be on your class’s spell list (or have otherwise been added to your spell list) and you must have a high enough key ability score to cast it.
Emphasis mine. Connection Inkling and Technomantic Dabbler may make you a spellcaster, but they do not grant you access to the Mystic or Technomancer spell lists because the feats do not make you a member of those classes.
I would say, however, that the bolded parenthetical implies that a character with either of those feats could use spell gems containing the spells that he knows from the feat without it counting against his daily limit.