Get an xml backup of the JIRA you are trying to import from.
Create the new project you are going to import things into.
Use Admin -> project import to extract the project issues from the backup xml into the new project with all comments, history etc.
There is no project export. Your question was about importing, not exporting, so I could not know what you were really trying to do.
Your options for that are:
1. Import the project you need into an empty JIRA (you'll need to manually recreate all the fields and config. OR import the whole thing into an empty JIRA and delete the projects you don't want to share. Then export that and give it to your client
2. Find add-ons that can export projects
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