Can I encode a graph in a unary alphabet
As you mention in your question, everything that can be coded in binary (that is, every countable set) can also be encoded in unary. Arrange all binary strings in some order, say
$$ \epsilon, 0, 1, 00, 01, 10, 11, \ldots. $$
Let $w_i$ be the $i$th string in this order. You can convert from binary to unary by mapping $w_i$ to $1^i$ and vice versa.

You mention that you are worried that this encoding is not efficient. From the perspective of complexity theory, this is not so clear, since if your representation is very large, then it is fair to allow you a lot of time to encode and decode it. In formal terms, for a representation of length $n$, you should be able to encode and decode in time polynomial in $n$.

Consider the example of graphs. In order to encode a graph in unary, you first encode it in binary, say using $m$ bits. Then you output $1^n$ for some $n \approx 2^m$. You can easily compute the output in time polynomial in $n$. Going the other way around, you can convert $1^n$ to the corresponding binary string $w_n$ in time polynomial in $n$ as well, and decoding the binary representation only takes time polynomial in $m = O(\log n)$.