Do galaxy mergers have a major impact on star formation rates within the galaxies
At first, the answer to your question seems straightforward. When galaxies merge, the gas contained in them gets swirled around, which can make the gas collapse and form new stars much quicker than usual. There are some galaxies we know of that are currently merging and have very high star formation rates. For example, the Antennae Galaxies (see image) are two merging galaxies that are currently undergoing a rapid formation of stars.
credit: NASA/ESA Hubble Space Telescope
However, in the last few years, with the rise of large galaxy surveys, we have learned that the picture is a little more complicated. When averaging hundreds of thousands of galaxies, we find that the difference in star formation between merging and not merging galaxies is minor, something like a 20% increase. The short explanation for this counterintuitive result is that while galaxies with very high star formation rates are often mergers, not ALL mergers display this behavior. Multiple additional factors decide whether or not a large amount of new stars is formed. How much gas the merging galaxies contain is one of them. When much of the gas has already collapsed, e.g. during previous merging events not much is left to form new stars. For very old and large galaxies, we sometimes even see less star formation in merging galaxies. From this, it is intuitive to see that when we look at very distant merging galaxies (they are younger because of the finite speed of light), we see a larger increase in the star formation rate. There are also arguments that the star formation rate is only increased for a small fraction of the time the merging of two galaxies takes and maybe there are only certain types of mergers where star formation is increased dramatically.
Overall I think that it is fair to say yes, galaxy mergers have an impact on the star formation rate of galaxies, but it is more nuanced than we have thought for a long time. For this reason, there is a lot of ongoing research on this topic and our understanding will surely improve further over the next few years.
More information see, for example, the introduction in Pearson et al. 2019 ( https://arxiv.org/abs/1908.10115)