= Solution
A <reduced positive definite binary quadratic form> $ax^2+bxy+cy^2$ satisfies $|b|\leq a\leq c$, with $b\geq0$ when $|b|=a$ or $a=c$. These inequalities select representatives under <proper equivalence of binary quadratic forms>; the changes of variables have integer entries and <determinant> $1$.
For the first <binary quadratic form>, put $x=u-v$, $y=v$. For the second, put $x=-u-3v$, $y=u+2v$. Both changes preserve <coprime integers>, and give
$$
f=3u^2+2uv+4v^2=3(u+v/3)^2+\frac{11}{3}v^2,\qquad g=u^2+11v^2.
$$
Proper representation means using $\gcd(u,v)=1$. For $f$, $|v|\geq2$ gives a value greater than $5$, while $v=0$ forces $u=\pm1$. For $|v|=1$, the smallest values are $4$ and $5$. For $g$, $v=0$ again gives only $1$ primitively; $|v|=1$ gives $11,12,15,\ldots$, and $|v|\geq2$ gives at least $44$. Thus \b[the three smallest properly represented values] are
$$
\boxed{f:3,4,5;\qquad g:1,11,12.}
$$
For example the original variables can be chosen as $(1,0),(-1,1),(-2,1)$ for $f$, and $(-1,1),(-3,2),(-4,3)$ for $g$.
Every positive definite <binary quadratic form> can be reduced: integer shears first make $|b|\leq a$, and interchanging the variables when $c<a$ decreases the positive leading coefficient, so this procedure terminates. To enumerate the <reduced forms of discriminant minus forty-four>, reduction gives $3a^2\leq44$, hence $1\leq a\leq3$. Checking $b^2-4ac=-44$ with the reduced inequalities gives exactly
$$
(1,0,11),\quad(2,2,6),\quad(3,2,4),\quad(3,-2,4).
$$
The second represents only even integers. The last two have the same represented values, by $u\mapsto-u$. The first and third are the reduced versions of $g$ and $f$. Therefore \b[every represented odd integer occurs in at least one of $f,g$], whether or not its original representation was primitive.
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