I think its very hard to draw conclusions. The country with the lowest rate is majority Christian and AFAIK is not extremely conservative: https://en.wikipedia.org/wiki/List_of_countries_by_suicide_r...

A second list on that page gives different numbers: https://en.wikipedia.org/wiki/List_of_countries_by_suicide_r...

In between is a table that shows the suicide rate is much lower in "upper middle income" countries than in either poorer or richer countries. I suspect that is a pointer. Affluent enough not to be constantly struggling, but different in some ways (urbanisation? materialism? isolation?) from the rich countries.

You also need to look at other factors. Conservative countries have younger populations, so you need to correct for that. Not all countries have equally accurate statistics.

I think conservative countries may have an advantage because of larger families, extended families, fewer people being single (and less isolated if they are) etc. but it needs more data. There are some pretty conservative countries with high rates.