...no-arbitrage models (such as Black-Scholes and HJM) are equivalent to
equilibrium models (such as CAPM or C-CAPM).
Short Answer Yes, for models where asset prices are assumed to be Ito semimartingales (where the martingale part is a Brownian integral), although a more general argument is needed than that suggested by the special cases typically encountered in finance.
Clearly, no-arbitrage is a necessary condition for general equilibrium.
To claim equivalence is then to say it is also sufficient, i.e. given any price process $P_t$ and a density process $D_t$ such that the discounted price $e^{-rt} P_t$
is a martingale after change of measure by $D_t$, one needs to find
a (say, representative) investor $u$ and a equilibrium consumption process $c_t$ such that
$$
e^{-\beta t} u'(c_t) = \lambda e^{-rt} D_t, \quad\quad (*)
$$
for some $\lambda > 0$.
In other words, one needs a "SDF/marginal utility representation" for equivalent martingale measure densities.
The equation $(*)$ is the usual relationship
$$
\mbox{marginal utility} \; \propto \mbox{price}.
$$
If you write down a heuristic Lagrangian, $\lambda$ is the Lagrange multiplier in the FOC.
In general, FOC's of this type is only necessary for optimality of $c_t$.
If $(*)$ is sufficient for optimality of $c_t$, you can take the dividend to be $c_t$ in $(*)$ and $P_t$ becomes the equilibrium price faced by the representative investor $u$.
With certain assumptions on $u$---such as concavity and the Inada condition,
Karatzas, Lehoczky, Shreve (1987) showed that this can be done when $P_t$ is an Ito semimartingale and market is complete. (See also Cox and Huang (1989).) The rigorous argument makes use of convex duality and is referred to as the martingale duality method in mathematical finance.
The Ito semimartingale case certainly covers many---perhaps most--- asset pricing models in finance.
In fact, asset prices are usually assumed to follow a very special Ito semimartingale---geometric Brownian motion, where the risk neutral density $D_t$ is itself an exponential martingale.
Then $(*)$ takes the special form
$$
\frac{dM}{M} = - r dt + \frac{dD}{D},
$$
where $M$ is the SDF.
One can then take the representative investor to be CRRA and, since powers of exponential martingales are still exponential martingales, back out an SDF $M$ without referring to a more general argument.
For example, as you already point out,
the Black-Scholes formula for pricing European call option in complete markets can be recovered from the Lucas asset pricing model, where the equilibrium cum-dividend return process follows
$$
\frac{d P + D dt}{P} = (\mu + \delta) dt + \sigma dW
$$
with $\delta$ being the endogenous dividend-price ratio $\frac{D}{P}$.
The underlying for the call being priced is the Lucas tree.
I don't know whether the martingale duality method has been extended to general semimartingales.
When market is incomplete, after some browsing it appears that only the case of terminal utility has been investigated and it is shown that certain constraints must be placed on $u$.
