Both of them can be translated as "question" in Japanese, and in many cases, they are interchangeable.
問いis a noun as well as 質問 is。It's a noun derivation from a verb '問う.'
As Ken-san answered above, 問い is not commonly used in conversation, while 質問 can be used in writing or speaking.
In my opinion, you make 質問 to specific someone or some people, expecting to get the answer. You may have a question(質問がある), make a question(質問をする), or may answer to a question(質問に答える).
問い is used to refer to the actual questions in exams. See the example#5 below.
Also, usually in writing, 問い is used to refer to philosophical or social questions. In this case, 問い is made not to make you answer, but to make you think.
例
記者会見で、イチローは「好きな食べ物は?」という○質問/○問いに「カレーです。」と答えた。
In the press interview, Ichiro answered "Curry rice." when he was asked what his favorite food was.
先生「○質問/×問いがある人は手を挙げてください。」
teacher "Please raise your hand if you have any questions."
italkiのQ&A掲示板に○質問/×問いを投稿した。
I posted a question on the Q&A board on italki.
よくある○質問/×問い
Frequently Asked Questions
日本語テスト
以下の○問い/△質問に日本語で答えなさい。
問い1. ........
問い2. ........
Japanese test
Answer the following questions in Japanese.
Q1.....
Q2......
人間とは何か?という○問い/△質問に正しい答えはない。
There is no answer for the question "What is human being?"