Yes, but not by default. You can embed any control you like in any DataGridViewCell but there are only certain embedded controls available by default and they only expose certain of their properties. Your two main options are:
1. Handle the EditingControlShowing event of your grid and, if it is in the appropriate column, get a reference to the TextBox that is the EditingControl and set its MultiLine property to True.
2. Inherit the DataGridViewTextBoxColumn and DataGridViewTextBoxCell classes and propagate the MultiLine property of the cell's embedded control up through the cell and column so you can set it at the column level.
The second option is more complex but will produce a better result. You can also implement functionality to display the TextBox at a larger size while editing to display more of the text. If you're inexperienced, though, you might like to stick with the first option for now. You won't actually see multiple lines unless you make the rows taller though.