Yes, generally the conversion of a convertible note does not result in a realized gain or loss and is not a taxable gain. There may be exceptions if Certain adjustments to the conversion ratio are made. This is because the adjustments may have the effect of increasing a convertible note holder's proportionate interest. This can result in a deemed distribution to the convertible note holder even though no cash is actually distributed.
The purchase of convertible debt is not a taxable event to the holder unless he/she transfers appreciated or depreciated property in exchange for the debt.