Here's a heuristic that suggests why arbitrarily large strings of consecutive 
numbers should be representable by some binary quadratic form.  For simplicity 
consider a prime $p$ that is $3\pmod 4$ so that $-p$ is a fundamental discriminant.  Suppose that $p$ has been chosen in such a way that all the primes $\le k$ are 
quadratic residues $\pmod p$.  Suppose there are $h$ quadratic forms of discriminant $-p$ (and note that there is only one genus).  

Recall that a number $n$ is represented by some form of discriminant $-p$ if 
every prime factor $\ell$ of $n$ satisfies $\chi(\ell)=1$.  As discussed in my 
answer to https://mathoverflow.net/questions/29280/achieving-consecutive-integers-as-norms-from-a-quadratic-field we should expect to find many strings of $k$ consecutive numbers, each of which is representable by some form of discriminant $-p$.  

In my answer to that question, I focused on such strings of (almost) prime numbers, 
but the same Hardy-Littlewood heuristics would predict lots of strings $n+1$, $\ldots$, 
$n+k$ where each $n+j$ is divisible only by primes that are quadratic residues 
$\pmod p$, and each $n+j$ has a typical number of prime factors.  Under the 
restriction that all prime factors of $n$ are quadratic residues $\pmod p$, if $n$ is 
large then typically it will have about $\frac 12 \log \log n$ such prime factors.  Moreover we may expect these prime factors to be roughly equally distributed 
in the class group (which is of fixed size $h$).  Thus since there are $2^{\omega(n)}$ 
factorizations of $n$ as a product of two ideals, we would expect that typically there are many such factorizations with one of the factors lying in a prescribed 
ideal class.  

Summarizing one would expect that there are many strings $n+j$ 
($1\le j\le k$) with each $n+j$ composed of about $\frac 12\log \log n$ primes 
that are all quadratic residues $\pmod p$, and (typically) each such $n+j$ would 
be represented by every form of discriminant $-p$.  

It should be possible with a little effort to turn this into a precise 
Hardy-Littlewood type conjecture, but I don't see any hope of a proof. 

The heuristic described above is probably classical.  One place where this 
heuristic is described is a paper of Blomer and Granville: see pages 9 and 10 of 
http://www.dms.umontreal.ca/~andrew/PDF/quadraticforms.pdf