The short answer is no; the detailed answer depends entirely on what is
meant by "explode."  The Sun doesn't have anything like enough mass to
form a Type 2 supernova (whose progenitors are supergiants), which
require more than about 8 solar masses; thus the Sun will not become a
supernova on its own.

"Novae" arise from an accumulation of gases on a collapsed object,
such as a white dwarf or a neutron star.  The gas comes from a nearby
companion (usually a distended giant).  Although nova explosions are
large by human standards, they are not nearly powerful enough to
destroy the star involved; indeed, most novae are thought to explode
repeatedly on time scales of years to millenia.  Since the Sun is not
a collapsed object, nor does it have a companion---let alone a
collapsed one---the Sun cannot go (or even be involved in) a nova.

Under conditions not well understood, the accumulation of gases on a
collapsed object may produce a Type 1 supernova instead of an ordinary
nova.  This is similar in principle to a nova explosion but much larger;
the star involved is thought to be completely destroyed.  The Sun will
not be involved in this type of explosion for the same reasons it will
not become a nova.

When the Sun evolves from a red giant to a white dwarf, it will shed its
atmosphere and form a planetary nebula; but this emission could not
really be considered an explosion.