Is heat the same as internal energy? If not, give an example in which as system’s internal energy changes without a flow of heat across the system’s boundary.

No, heat is not same as internal energy.
Energy possessed by the atoms or molecules by virtue of their motion is called kinetic energy. There is a force of attraction between any two molecules; so they possess some potential energy. Sum of total kinetic and potential energies of atoms or molecules constituting a system are called the internal energy of the system. Internal energy of a system depends upon the parameters of the system. It has definite value for a definite thermodynamic state and is not a measureable quantity.
When a cold body ‘A’ is placed in contact with a hot body ‘B’ something is transferred from hot body ‘B’ to the cold body ‘A’ which results in a rise in temperature of the cold body. This transference stops when the two have acquired same temperature. This indicates that only a part of the energy of ‘B’ is transferred to ‘A’. This part is called heat.
Thus heat is the part of internal energy which is transferred from one body to another an account of the temperature difference.