It will open a gate.
The key language is "similar."
The referenced description states:
Placing the haversack inside an extradimensional space created by a bag of holding, portable hole, or similar item instantly destroys both items and opens a gate to the Astral Plane.
The first three items, the Bag of Holding, Heward's Haversack, and the Portable Hole are all variations on a theme. So the question is if the Quiver is similar to them:
- Like all three the Quiver of Ehlonna is bigger on the inside.
- Like all three the Quiver of Ehlonna can fit many items without ever changing its outer dimensions.
- Like all three the Quiver of Ehlonna can fit many items without ever changing its weight.
- Like the haversack the Quiver of Ehlonna allows you to simply pull out what you need without rooting around for it.
- Like all three the Quiver of Ehlonna is specifically described or referenced as using extradimensional space.
Considering that even the haversack, the hole and the bag have some differences I would rule that the quiver has enough similarities to be considered similar to these three items, and especially so because it is described as using extradimensional space. Hence, placing the Quiver in any of the others would open a gate to the Astral Plane.
Left unsaid is what happens if the quiver is ruptured. The bag and the haversack each have different consequences for being ruptured while the hole and the quiver make no mention of it.