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In a recent game my Inquisitor Warlord rolled the trait where I get to "select one of the Warlord’s ranged weapons. The weapon gains +1 Strength and the Rending special rule for the duration of the battle." I had planned on her manning my Quad Gun as long as it was operable.
Though I didn't play it this way, could she have given the Quad Gun the +1Str and Rending, or does it have to be one of her "personal" weapons?

I've seen some liberties taken with the Quad Gun depending on what character is manning the gun (ie, Vindicare Assassin) is this any different?

Negative, due to a timing constraint.

This warlord trait is evoked at the very beginning of the game, once you roll it as the result or it is selected for you as per the Warlord Trait rules. It contains the restriction that the weapon you are selecting must be one of the Warlords weapons, requiring the weapon to be listed as part of the commanders wargear at the time this rule comes into effect. As the Quad Gun is not listed under the Warlord's wargear as a weapon they possess, and the rule granting you permission to treat it as such can only be evoked when it is fired, it does not meet the requirements to be considered a 'personal weapon' during the pre-game setup.

Those 'liberties' as you call it, can be chalked up to a difference in the wording of their respective rules. to provide you with an answer: no you pay not apply that warlord trait to a quadgun.

Some of the vindicares rules refer to the vindicares 'shooting attacks' (aka: anything the vindicare can be said to be shooting with regardless of source), whereas the rule you are wondering about talks about the inquisitors 'ranged weapons' (ie: those listed in his or her profile.)