Answer yes absolutely the quality of the UPS can alter battery life.
Any quality UPS should do a periodic self-test. For APC kit it will normally click and clunk for 10-15 seconds every 2 weeks from when it was powered on. A UPS that does this should get 5-6 years off its batteries before they need replacing.
A budget UPS will not do these tests itself. Some can be forced to do a battery test by software, but that relies on the user setting up said monitoring software and connecting a serial/USB cable.
Secondly, a good UPS will have a fan for when the inverter is operating. A budget UPS tends to skimp.
So a budget UPS with 1-2 12V batteries will get about 2 years.
HOWEVER the budget UPS won't tell you when the batteries are expired. Your first clue might be a tiny flick in the power, enough to make the UPS jump in but not enough to upset non-UPSed gear.
The UPS makes a leap onto battery, but it finds theres nothing in the battery, so it turns straight off and takes the load with it. The UPS can generally be turned back on and services resume, but its only function is trim of overvoltages. In this state, it has no reserves and can't boost a low voltage, and can't carry the whole lot in the event of no utility.
Example "budget" UPS (this model is rebranded by multiple companies, and retails for about $100)
By comparison, a quality UPS is a brand name, and if new costs 10-20 times as much as the budget one. Your requirements should dictate what kind of UPS you choose - sometimes a budget one is the right thing to have.