Is there any increase in legendary drop rate with the True Vault Hunter mode or the Vault Hunter's Relic?

Vault Hunter's Relic

It has a very minor effect on legendary drop rates, that you're probably better off using another relic. It only affects white (common) and non-white drop rates (and not legendary drop rates, specifically).

From this somethingawful.com forum post by user, gibbed:

So I just investigated the Vault Hunter Relic data.

It modifies two attributes:

GD_Balance.Weighting.GearDrops_RareWeightModifier: +0.05 GD_Balance.Weighting.GearDrops_CommonWeightModifier: -0.06

Let's set them to +100.00 and -100.00!

Net result: I have not seen a single white weapon drop, greens drop, seen a few blues. Nothing else.

So, what I can gauge from this, is that it shifts a 5% of your chance for whites into non-whites. Doesn't touch anything else. Very low impact, perhaps bordering on useless.

True Vault Hunter Mode

Some of differences in True Vault Hunter Mode as opposed to Normal mode are:
...

• Better Loot: Due to the nature of difficulty increase and the fact the game becomes more difficult, in True Vault Hunter Mode there is a higher chance of enemies dropping better loot and chest also will contain better loot.
• Better Loot more frequently: Enemies in True Vault Hunter Mode will be have a higher percentage chance of dropping better loot as they have more health and are more difficult to defeat.

^{Source: Borderlands wiki, "True Vault Hunter Mode" article}