çiçek , yes, means flower but çiçekği is not a correct form of writing (i.e. incorrect).
you should write it as "çiçeği"
this is caused by one of basic rule of turkish grammar.
and yes, çiçek can also be used for a girl name.
the clearer difference between çiçek and çiçeği is as given below
çiçek --->> flower
çiçeği --->> as in a sentence like: "I bring/brought this flower."
Pay attention to the word forms "çiçek" and "çiçeği". They are different forms. What happens here is called Final-obstruent devoicing.
Simply put, when a vowel initial suffix is added to a word ending in one of “p – ç – t – k” consonants in Turkish, the consonant (“p – ç – t – k”) turns into “b - c - d - g”, respectively. Therefore, çiçeK --> çiçeĞi.
“-i”, on the other hand, is the accusative ending in Turkish (also –ı, -u, -ü according to the vowel harmony).