Is it safe to get a relaxer in first trimester?

Probably. There is no data that suggests it's unsafe, assuming you don't drink the relaxer, but it makes sense to avoid toxic chemicals in all respects. As for hair coloring, versus relaxing, henna-based colors are harmless.
Wait. It is recommended that any type of hair treatments, including coloring, relaxers, etc, be saved until further in the pregnancy. This just follows the old adage that "it is better to be safe than sorry" due to the chemicals having potential to cause problems.
Probably. Hair relaxers, hair dyes, permanents all require chemical exposure. Most likely there is very minimal exposure throughout your system making this safe. There are no known specific fetal risks associated with pregnancy. That being said many pregnant women chose to postpone these treatments until the 2nd trimester when most fetal organ development is complete.