You need to work with the brokerage that handles the stock plan to make any transfers.

Gifting stock to your spouse is not a taxable event.   Contact the broker to move the stock or retitle the account. 

When a married couple is filing joint and transfers assets from one spouse to the other, that is not a taxable or tax reportable event. At least, not on the federal taxes. There may be exceptions. But I'm not aware of any. As for state taxes,I don't know enough about state taxes among the 37-plus states that tax personal income to make any calls on that.

The filing status of a married couple is irrelevant for determining whether or not a gift tax return (Form 709) is required to be filed for transfers between spouses.