I've heard the argument before (temporary idea and effective means), which probably is true of the non-terminally ill. I haven't seen any data supporting it so far. I've also haven't seen any definitive evidence against it, only suggestive stuff.
There are some states with restrictive gun laws that show a slight decrease, but others seem to be unaffected (continue to increase). Just as there were less restrictive states that show the same mixed behavior. (CDC data referenced by both sides)
One thing that I saw in the CDC data was that suicide rates have risen over the past 30 or so years for all states. Non-firearm suicides are 27% more common since the 80s, but it didn't look at why firearms are decreasing as a percentage. RAND says the number of homes with at least one firearm has stayed fairly consistent over that time (40% -/+ 5%). So it seems that there isn't a substantial reduction in suicides based solely on gun control laws, but it seems it's also impossible to say if the number would have been higher without those restrictions. The fact the ownership has stayed the same, but firearms are a lower percentage of suicide cases could just be demographic or preference issue, but we don't know.
One interesting thing I observed which I didn't see studies about (because both sides are focused on just confirming their positions on pro/anti gun policies) is the correlation between mental healthcare service ratings and suicide. This is just rough, but I found these two maps appearing to be more correlated than the gun ownership and suicide maps/studies. Just something I found interesting and would need an actual study to look into it.
https://www.mhanational.org/issues/ranking-states
https://www.cdc.gov/nchs/pressroom/sosmap/suicide-mortality/...