I noticed lots of compound words using "只" for "only", but what is the correct way to say "only child" (a child with no brothers, nor sisters)?
The Chinese expression for 'only Child' can be '独生子女(only child)','独子(only son)','独女(only daughter)'，or some more ancient or literary words like '独苗(only child)','单丁(only son)',单根独苗(only child or only descendant).
Native Chinese speakers may have other expressions like,
他膝下仅有一女。（He has only one daughter.）
他是那老汉唯一的儿子。（He is the only son of that old man.）
他是三代单传。（He is the only one son for several generations in the family.）
I hope the above helps.
There's an exact word for
独生子(女), especially in China mainland.
根据family planning policy（计划生育政策），一般情况下每个家庭只允许生育一个孩子，也就是我们所说的only/single child（独生子女）。
According to the context, there're also other expressions. From youdao:
If you have an only child, was this by choice or chance?
After counting down for nine years, I was finally an only child.