Yes, there was a beginning to time.  However, a cautionary note is in order.  Although our psychological sense of past, present, and future is strong, time may not be a fundamental physical entity at all.  Rather, it may simply express the numerical order of material changes.

For example, when a photon of light moves from position 1 to position 2, position 1 doesn't exist before position 2 in time, only in numerical order.

In this sense, the universe is timeless even though we use the word "spacetime" to describe it.  The universe isn't 3D in space and 1D in time, but rather 4D in space.

Time plays an important role in comparing durations and in describing physical systems but it may describe only how space is quantum-networked and entangled, rather than describing something apart from space.

That being said, our real time began when our universe did, in an inconceivably hot, dense, tiny spot just after the "Big Bang".  The bang wasn't an explosion in the usual sense of the word but rather the sudden appearance of expanding space.  Since then, a duration equal to about 13.82 billion Earth years has elapsed.

"Big Bang" is a term coined somewhat mockingly by Fred Hoyle, a 20th century English astronomer, science fiction writer, and big-bang denier.  In 1948, he argued against a beginning of time, proposing instead a "steady state" universe with infinite time.  He rejected both the Big Bang theory and the Book of Genesis as pseudoscience that suggests a creator.

But the challenge for Hoyle's theory was the observed expansion of our universe over time, hardly a steady state.  To explain the expansion, confirmed in 1927 by Edwin Hubble, Hoyle postulated Mini Bangs in between the galaxies that keep the outward flow of matter constant, like a steadily flowing river.

But Hoyle's job got more difficult in the early 1960s when radio sources were found to be more prevalent in the early (far away) universe than in more recent times, once again contradicting a steady-state universe.  In 1963, quasars, which almost only occur in the distant past, were discovered.

Then, in 1965, the observation of the cosmic microwave background (CMB), a radiation with no discernable source, pretty much killed the Steady State theory.  What's more, the Big Bang theory had predicted just such a background radiation.

When physicists try to extrapolate back in time from the earliest well-understood state of the universe, a miniscule fraction of a second after the big bang, they quickly hit a gravitational (a curvature) "singularity" where classical laws break down:  gravity, material density, and temperature become infinite and spatial dimensions shrink to zero.

To the rescue comes imaginary time, which treats time like a dimension of space and successfully predicts quantum behavior.  The state of our universe in real time can be calculated from the state of our universe in imaginary time.

Moreover, singularities like the big bang don't interrupt the flow of imaginary time.  Therefore, a beginning to real time doesn't violate any physical laws or call for a creator.

Steven Hawking has shown that the Big Bang could've been an ordinary point on the 3-dimensional surface of spacetime just like the North Pole is an ordinary point on the 2-dimensional surface of Earth.

The sudden expansion of such a singular point and the rapid inflation of the universe could borrow energy from the intense gravitational field to create matter.

In imaginary time, no physical laws would be violated and the gravitational debt could be repaid at the end of the universe.

In 1994, when a Big Bang renaming contest in Sky and Telescope magazine failed to pick a winner,  Fred Hoyle said of the Big Bang, "Words are like harpoons...Once they go in, they're very hard to pull out."