In contrast to English which has two different words "happy" and "lucky" German has only one word of neutral gender: Das Glück. In German happiness is seen as part of luck. If you want to show the difference in meaning the verb after Glück can help to disambiguate:
Glück haben means be lucky (noun + verb to have)
Glücklich sein means to be happy (adjective + to be)
The English word luck is related to German g(e)lücke,"fortune, good luck." Probably borrowed in English as a gambling term.
The adjective always refers to happines as happiness is seen as a quality whereas the noun refers to luck. There are some nouns whcih refer to happiness like Freude or Freulichkeit but they are semantically a bit different. Freude might be more forceful than happiness and Freulichkeit may refer to merriness.
In addition German has a lot of word combinations like Viel Glück: all the best, Glückspilz: lucky devil, Glück im Unglück: a blessing in disguise, Glückwunsche: congratulations
Often the word "Schwein" pig related to English swine iinformally replaces Glück: Schwein gehabt: I was lucky which can also mean it was a narrow escape.