Disclaimer: This answer was written before the addition of the 5e tag. The content below may not be applicable to that specific edition of D&D.
D&D 3.5's Draconomicon actually has several examples of dragons having specific smells. It seems to vary by dragonflight color:
Black - Smells like rotten vegetation and foul water.
Blue - Smells like ozone and sand when upset.
Brass - Smells like hot metal and desert sand.
Bronze - The smell of the sea.
Copper - a "stony" odor.
Gold - Safron and incense.
Green - Chlorine.
Red - Sulfur and pumice.
Silver - The scent of
plot devices rain.
White - Couldn't find anything about its scent.
From the classic species, none of them has the smell of lilac. More so, from what is possible to see, it seems that every species of dragon has a different scent, so I would say that no - in general, dragons don't smell like lilac.
Keep in mind: Those are the scents for their regular, dragon form. I wasn't able to find anything regarding their scent either changing or remaining the same when turning into a human.
A bit extra: I was curious about a possible link between lilacs and dragons, and I went on the web to search more about it. There seems to be quite a few references of both "lilac" and "dragon" appearing together, but most of them are irrelevant to this context. There seems to be some fanfic regarding either the species of "lilac dragons" or a character named "Lilac", which seems to be a dragon, from the game Freedom Planet.
Most of those links were also tagged with "Sonic", "Romantic", or "Vampire" so I was somewhat scared to click anything.
Bad things usually happen when you follow links to fanfic sites.
Very bad things.
Strange fanfics aside, I was able to find a mild link between dragons and the the scent of lilacs on the series Ultimate Magic, by T.A. Barron - The dragon Basilgarrad seems fond of that scent and he seems to be able to produce their smell out of thin air at a given moment. I don't own the books and I was able to check just an excerpt of it, so I don't know the relevance of this.
The Lavender Dragon, by Eden Phillpotts, has a dragon which smells (suprisingly) of lavender. Not quite the same as lilacs, but it somewhat fits the theme.
Flight Rising has a relatively large thread dedicated to what each user's dragon would smell like. This thread contains a lot of very different designs for dragons, going from traditional ones to things that don't look at all with a dragon, but is a very interesting read with a lot of pretty pictures. Very colorful regarding descriptions of why each dragon smells like they do.
Caught By the Dragon: Maiden to the Dragon by Mac Flynn features a very... off-brand novel about a shapeshifter dragon that creates the perfume of lilac to seduce a female human. This one was catch up by @A C.
Some Extra Thoughts
After writing this answer, I began to take more notice of references to scents, colors and flowers in the fantasy material I read. There seems to be an extreme tendency to relate the color purple, in different shades, to magic - in special, references to both "Arcane" and "Eldritch" magic, in several different materials, tend to push towards the color purple. TVTropes seems to agree with my observations.
My theory is that since people relate the color purple to "magical stuff", the usage of a purple scent eventually bled out to literature that deals with magical beings of some sort. Since Dragons seem to be one of the most popular fantasy creatures on fiction, that association eventually came up naturally by sheer statistical pressure.
I'll keep an eye out for more flowery-scented, purple-colored, dragon-related content, and update this answer again if I end up finding something else interesting.