There is no apparent relation.
The English name Chechnya is a transliteration of the Russian (Чечня́), which according to Amjad Jaimoukha's The Chechens is a 17th-century Russification of Shashan, a Kabardian name for the people who call themselves Nokhchiy (which may be derived from the words for sheep cheese, the plow, or the Biblical Noah). The traditional etymology links it to the village of Chechen-aul, where the Russians were defeated in battle, but the name predates the battle.
The Czech Republic takes its name via Polish from the Czech tribe, which conquered the region in the 9th century. That name in turn may derive ultimately from četa, a military unit in several West Slavic languages. It dates from the 18th century; the traditional English name was Bohemia, used into the 20th century. The traditional etymology links Czech to Čech, the legendary founder of the country.