The only kind of a battery that affects the time of your watch is an empty battery. As long as it within a certain range of voltage, it makes absolutely no difference. The watch runs or it doesn't.
What concerns the possibilty to regulate a quartz watch, there is nothing much toi do except changing the quartz unit.
The quartz watch (as it has already been said before) is a step forward in precision over the mechanical watches, keeping better time at $ 9.99 than most of the $ 75,000 watches. If it's seconds a week, that's the limit.
You can buy one of the atomic watches now also available as wrist watches for $ 12,000 and you have the time kept within 1 second +/- in 1000 years.
On a larger scale, the atomic clocks are within +/- 1 second in 40 million years, and that is already in conflict with the changing rotation of the earth which creates the needs for adjustments once in a while.
If you really worry about a few seconds a month, you will not get this settled with a quartz watch. You need a radio-controlled watch, which is a regular quartz watch taking a time signal from a big atomic clock at certain intervals, which frequently regulate the time - the perfect time. That's the end of time keeping on your wrist. Do not go for a cheap radio-controlled watch which is as accurate execpt that the hands are falling off sooner or later. Junghans had invented that thing and is somewhat on the top of the price scale. You can go down to a Citizen watch, and then to Casio, which is at the lower pricing end in the quality segment. There are others which someone might recommend.
I have a Casio radio-contolled with solar power. No more batteries and always the perfect time. Display analog and digital, date of course, other gimmicks I don't use. It goes to sleep if put in a drawer and can live without light for several months. And last night, it switched to summertime automatically. With that thing on my wrist, I now enjoy my watch collection much better than before. I see that the watches are running well and don't worry about minor deviations. Before that, I had all of them (mechanical) on the time-scale with endless attempts to get them better regulated, which was good for one stable position only at a certain temperature. And yes, I also wanted to 'regulate' my quartz watches before, now it's more keeping them nice and need.