レッスン時、出来たら会話を録音したいのですがいいですか?って英語でなんて言うの？

"May I record our lesson?"
May I - You are seeking permission to record the lesson.

"Is it okay to record the lesson to review later? "
Is it okay to - this is another way to ask permission to record. Then you go on to explain why you want to record the lesson.
Review later - To listen back and hear what you learned before.

Do you mind - This is asking someone if they have any objection.
Example:
Q: Do you mind if I record this lesson?
A: That is no problem.

Is it okay if I record this lesson? - This is asking for permission to record the lesson.
Q: Is it okay if I record this lesson?
A: Sure, go ahead.

To ''go ahead'' means to proceed.

In this example, it is asking the specific wishes of that teacher because perhaps the company or another teacher would have a different answer. Also, by adding "today" to the end, it clarifies that you only want to record that lesson, not every lesson that you have.

I hope that this helps :)

Probably your teacher will applaud you recording and reviewing your lesson as it gives you a chance to review everything and prepare any questions you may have for the following lesson. It will be especially helpful when reviewing pronunciation of unfamiliar words.

Sometimes you might want to review the lesson later on to see how well you are doing or any mistakes it is always good to ask permission to record soemone else so if you want to ask if it's ok to do then you would say 'can I record the lesson?' or 'would it be ok if I recorded the lesson?'

Recording online lessons is a good idea if you would like to go back and review what was learnt.
You can also get to make more notes when you have the class recording.
Its great to just let the tutor know your recording, out of courtesy.

Review: look back on, look at again
Courtesy: politeness

"Teacher, do you mind of I record this lesson? I'd like to review it some more after the lesson."

"Can I record our lesson?" and "Is it okay if I record our class?"
These questions are politely asking if you are able to 'record'/video a class/lesson so you are able to access it for future purposes.
"Is it okay?" is a very casual expression for asking 'Can I?', 'Am I able to' etc.