What a neat question!
The answer to your stated question is Yes, you would be able to detect it, but not for the reason you might suppose.
The Beholder's cone behaves as the spell antimagic field, with a few exceptions (shape, range, and at-will-ness). Since this cone is an area effect spell, it follows those rules. Specifically, PHB 204 says
A spell's effect extends in a straight line from its point of origin. If no unblocked straight line extends from the point of origin to a location within the area of effect, that location isn't included in the spell's area. To block one of these imaginary lines, an obstruction must provide total cover, as explained in Chapter 9.
And it turns out that your simple wooden door does, in fact, provide cover. As such, the Antimagic Cone doesn't penetrate the door, and our daring hero is not affected by the spell.
On the other side of the door, detect magic is a self-targeted spell. Since the wizard is not under any particular effects, such as that field, the spell goes off without a hitch. Note that because it's a self-targeted spell, it isn't bound by the area rules. Its rules are somewhat special.
you can use your action to see a faint aura around any visible creature or object in the area that bears magic
with the exception
The spell can penetrate most barriers, but it is blocked by 1 foot of stone, 1 inch of common metal, a thin sheet of lead, or 3 feet of wood or dirt.
This means the wizard can, in fact, sense through our (presumably less than 3 foot thick) wooden door, and could notice the antimagic area on the other side of it.
However, it didn't seem to be the question you meant to ask. So, let's assume the beholder's side of the door is in total darkness (to prevent the wizard from seeing it), the wizard had opened the door, and the wizard hadn't cast the spell.
This basically just changes the scenario to be the wizard unknowingly casting inside an antimagic field.
According to the description for antimagic field,
Within the sphere, spells can't be cast
That is, if you're already in the field, the spell will simply fail. It's unclear to me from this description what you'd notice as a caster.
Presumably, as the area is "divorced from the magical energy that suffuses the multiverse", and as arcane casters are particularly in tune with that energy, you'd notice something was wrong.
Let's go back to the original question, and open the door (for some reason).
If you already had detect magic up, you'd have noticed an Abjuration around the area. Then, upon entering, detect magic would be suppressed and you would no longer sense the area as magical (along with anything else magical you could sense).
Interestingly, I wasn't able to find anything in the PHB or DMG about what you do, or don't, sense when a spell you're concentrating on is either suppressed or falls on an invalid target. As such, I believe it would be a DM's judgement call.