No. 私もこれの質問がありましたis incorrect. 私も同じことが聞きたかったです（I also wanted to ask the same question） or 私も同じ疑問を持っていました（I had the same doubt） is natural(correct).

I'm also still learning Japanese, so I hope this is correct for you. 1) Instead of これの, you can use この, as in この質問(this question) これ is a noun on it's own, so これの質問 is kind of like "this's question". この is sort of like the adjective version of これ. 2) The position of も. Where this particle sits in the sentence will change the emphasis. If you say 私も, it feels more like "In addition to all these other people, I also have a question." If we move the particle after 質問, it feels more like "in addition to all these other questions, I also have this other question." 3)Past or non-past tense verb. Depending on the situation you may want to use the non-past tense あります instead. If you currently want someone to answer your question 質問があります。 If the opportunity already passed, and you had the question before 質問がありました。 4)One of my language partners used this phrase もう一つ（もうひとつ）"one more/another". This might be a good fit for your question. 先生：わかりますか。"Do you understand?" 学生：すみません、もう一つ質問があります　"Sorry, I still have one more question." **Someone please correct me if I got something wrong.

私もこの質問を聞きたかったです。