When transferring shares from one account to another will Cost Basis Transfer over?
Yes.
Use the transfer share transaction and it will carry over purchase date and cost basis.
While I agree with @miklk 's response that the transfer share will do what you want, I suggest trying it and seeing it for yourself.
Make a Quicken Backup of your Quicken data file, then do a transfer of a single stock/bond and inspect the results. If you don't like the results, do a Quicken Restore and you will be back to where you were before doing the testing.
