Can amide bonds fragment in ESI-MS?
Yes, amide bonds can fragment during electrospray ionization. 

Intentional fragmentation in a collision cell

Amide bonds are intentionally fragmented in mass spectrometers all the time, during proteomics analysis.  In that case, peptides are ionized, particular ions are isolated (by the quadrupole of the mass spectrometer), and then the intact peptide ions are fragmented intentionally in a collision cell inside the instrument.  After collision with high-energy gas molecules, peptides fragment in mostly predictable ways giving rise to y, b, a, and z ions.

"In-source" fragmentation

The key step of the fragmentation process is collision with high-energy gas molecules.  That can happen accidentally during electrospray ionization in the ion source.  That's because the ion source, being at atmospheric pressure, has many many more gas molecules around than a collision cell, which might be at 10$^{-5}$ torr or lower in pressure.  So after your ions of your peptidic molecule form in the ion source, some of them might bump into these atmospheric gas molecules and be fragmented in the ion source, before they even make it into the mass spectrometer.

Controlling in-source fragmentation

The propensity of this "accidental" fragmentation to happen is somewhat tunable by ion source parameters -- nebulizer and capillary voltages, gas flows and temperatures, etc.  The details depend entirely on your exact model of MS.  If the carboxyl and amine fragments are both more intense than the "molecular ion", it could be a sign that the ion source is tuned too aggressively -- trying lowering a voltage or two and see if you boost the molecular ion signal.  (Alternately, for some applications, in-source fragmentation could be desired.)