The principal invested via a convertible note should accrue interest, but the rate of interest should be lower than a typical non-convertible loan from an investor. The range I see is 2-8% for convertible notes. Lower on the coasts and internationally, higher in the middle of the country.
If a potential investor plays hardball and insists on a high interest rate (i.e., 8% or more), then you might have the wrong potential investor. Or it’s at least a yellow flag and you should pay attention more closely. Although, if your startup is raising a small convertible note round (sub $100k), a couple extra interest points may not be that material.
The convertible note interest should also accrue until the note converts into equity or is paid out at the sale of the company. When the convertible note converts into equity, the interest also typically converts into the preferred stock, rather than being paid out.
If your investor wants a quarterly payment for the convertible note interest then he or she should invest in blue chip dividend stocks or bonds. The purpose of interest in a convertible note is to technically qualify the note as a debt instrument, and less about appreciating the time value of money — and definitely not about paying the investor interest with his or her investment money.