source: this began by giving an answer at enter link description here
This turned out sneakier than I expected. I wound up parametrizing $ Q(a,b) = Q(c,d) $ by Gauss composition, in the Dirichlet version, as $(a,b) = (x,y) \circ (z,w)$ while
$(c,d) = (x,y) \circ (w,z).$
That is: with positive $a,b,c,d$ integers and $a^2 + 3ab + b^2 = c^2 + 3cd + d^2 ,$ we get an infinite number of examples from $$ a = xz-yw, \; \; \; \; b = xw + 3yw + yz , \; \; \; $$ $$ c = xw-yz, \; \; \; \; d = xz + 3yz + yw . \; \; \; $$
I think this gives all (doubly) primitive examples, as in both $\gcd(a,b)=1$ and $\gcd(c,d) = 1.$
That is the question, does my simple recipe give all primitive examples? Some references are in Barry Smith's article Barry Smith on Zagier's methods for binary quadratic forms... I know some things but have no complete proof For instance, it's true if the common value is prime.
NOTE: it is beginning to seem that this is a consequence of unique factorization in the ring of integers of $\mathbb Q [ \sqrt 5] .$ For comparison: in the rational integers, when some $AB=CD,$ we may define $q = \gcd(A,D)$ and fiddle a bit to conclude that $A = g \alpha, B = h \delta, C = h \alpha, D = g \delta.$
examples:
n = a^2 + 3 a b + b^2 = c^2 + 3 cd + d^2
n a b c d x y z w n
11 2 1 1 2 1 0 2 1 11 = 11
19 3 1 1 3 1 0 3 1 19 = 19
29 4 1 1 4 1 0 4 1 29 = 29
31 3 2 2 3 1 0 3 2 31 = 31
41 5 1 1 5 1 0 5 1 41 = 41
55 6 1 1 6 1 0 6 1 55 = 5 11
59 5 2 2 5 1 0 5 2 59 = 59
61 4 3 3 4 1 0 4 3 61 = 61
71 7 1 1 7 1 0 7 1 71 = 71
79 5 3 3 5 1 0 5 3 79 = 79
89 8 1 1 8 1 0 8 1 89 = 89
95 7 2 2 7 1 0 7 2 95 = 5 19
101 5 4 4 5 1 0 5 4 101 = 101
109 9 1 1 9 1 0 9 1 109 = 109
121 7 3 3 7 1 0 7 3 121 = 11^2
131 10 1 1 10 1 0 10 1 131 = 131
139 9 2 2 9 1 0 9 2 139 = 139
145 8 3 3 8 1 0 8 3 145 = 5 29
149 7 4 4 7 1 0 7 4 149 = 149
151 6 5 5 6 1 0 6 5 151 = 151
179 7 5 5 7 1 0 7 5 179 = 179
199 10 3 3 10 1 0 10 3 199 = 199
205 9 4 4 9 1 0 9 4 205 = 5 41
209 13 1 8 5 6 -1 2 1 209 = 11 19
209 8 5 5 8 1 0 8 5 209 = 11 19
211 7 6 6 7 1 0 7 6 211 = 211
241 9 5 5 9 1 0 9 5 241 = 241
281 8 7 7 8 1 0 8 7 281 = 281
319 15 2 9 7 7 -1 2 1 319 = 11 29
319 9 7 7 9 1 0 9 7 319 = 11 29
341 17 1 13 4 5 -1 3 2 341 = 11 31
359 10 7 7 10 1 0 10 7 359 = 359
361 9 8 8 9 1 0 9 8 361 = 19^2
451 10 9 9 10 1 0 10 9 451 = 11 41
451 17 3 10 9 8 -1 2 1 451 = 11 41
551 22 1 10 11 7 -1 3 1 551 = 19 29
589 20 3 15 7 6 -1 3 2 589 = 19 31
n a b c d x y z w n