Every time I play Thresh with Relic Shield 
and its upgrades 
i find that i cannot proc its passive executions properly. I have noticed that for me to properly "execute" the minions I would have to last hit said minion, instead of killing it at the health threshold.
This is not a bug. The Relic Shield line does not deal execute damage on ranged attacks; read the tooltip.
Thresh, despite having 2 different attack animations that trigger based on his range, is and always has been classified as a ranged champion. You can still get the gold effect (as you've already noticed), you just don't get the extra damage to assist you.
He is ranged. Only executes for Melee champs.
2 Gold Generation, 75 Health. UNIQUE Passive: Spoils of War: Melee basic attacks execute minions below 195 (+5 per level) Health. Killing a minion heals you and the nearest allied champion for 20 Health and grants them kill gold. Allies who benefit from Spoils of War also gain +1 to their minions killed score. These effects require a nearby allied champion. Recharges every 40 seconds. Max 2 charges.
Melee basic attacks execute Melee basic attacks execute Melee basic attacks execute
Thresh is ranged.
Thresh can't execute minions with the relic shield line due to his auto attack being considered a range aa.
