Fraud in england-and-wales is defined within the Fraud Act 2006. The operative type of Fraud in this case is defined by Fraud Act 2006 section 2 (Fraud by False Representation).
This says:
Fraud by false representation
(1)A person is in breach of this section if he—
(a)dishonestly makes a false representation, and
(b)intends, by making the representation—
(i)to make a gain for himself or another, or
(ii)to cause loss to another or to expose another to a risk of loss.
(2)A representation is false if—
(a)it is untrue or misleading, and
(b)the person making it knows that it is, or might be, untrue
or misleading.
(3)“Representation” means any representation as to fact or law,
including a representation as to the state of mind of—
(a)the person making the representation, or
(b)any other person.
(4)A representation may be express or implied.
(5)For the purposes of this section a representation may be regarded as
made if it (or anything implying it) is submitted in any form to any
system or device designed to receive, convey or respond to
communications (with or without human intervention).
Prima facie, yes, making any representation (in this case by way of a forged document or lie to the landlord) can be Fraud if it is dishonest, and was intended to make a gain for the person making the representation or a loss for another.
Let's suggest the potential potential lessee said "I earn 100k per year", but actually only earned 50k per year, and L said that in order to convince the landlord that L was more credit worthy than he actually was, that would (likely) be fraud.
However, there are a number of reasons why the behaviour described might not be caught.
Let us suggest a letting agency requires an unreasonable amount of information to let an apartment, requiring copious identification documents with a large number of arbitrary restrictions (sadly not unusual). The potential lessee (L) is able to provide all but one of these, but is unable to provide (e.g.) a second utility bill as the utilities were in the name of L's housemate. So L takes the PDF of a utility bill, invokes a PDF editor, and changes the name of his housemate to L's own name, thereby satisfying the agency's requirement. Has L committed fraud?
Re s(2), firstly, what is the representation? It might be argued that either the representation is simply that L is who he says he is (which is not a false representation), or that the a third party provided documentation shows that L says he is who he says he is (which is false). But in that case, a L is in fact who L says he is, it's unlikely L will cause loss to another (the letting agent or landlord), or a gain for himself, as the purpose of asking for the representation is presumably to establish that L is who he says he is. It might be argued in the alternative that even if L did make a gain or cause a loss, L did not intend that gain or loss, merely to satisfy the agency's requirements.
We'd also have to prove the representation was dishonest (s(1)(a)) in an R v Ghosh sense, meaning:
... a jury must first of all decide whether according to the ordinary standards of reasonable and honest people what was done was dishonest ... If it was dishonest ... then the jury must consider whether the defendant himself must have realised that what he was doing was by those standards dishonest.
The first leg is self-explanatory; I'd suggest there is a good chance of it failing. Re the second leg, it might be argued that L believed he was avoiding pointless bureaucracy, rather than being dishonest (compared to constructing an instrument to prove he was someone else).
I note in passing that the second leg may have been over-ruled in Ivey v Genting Casinos though it's not obvious to me whether that is just for civil cases given criminal cases normally require mens rea.
Finally, the question was asked in the context of the United Kingdom. The Fraud Act 2006 applies in england-and-wales and northern-ireland. The law in scotland is different.
