🚀 go-pugleaf

This answer is yes (it is decidable) for a large class of 
algebraic endomorphisms of varieties where everything is defined over the
rationals.  For example, let f(x) = x^2 + c where c is rational.
Define the height of a rational number p/q (where (p,q)=1) as
h(p/q) = log max(1,|p|,|q|).  Then it is easy to see that 
for x rational, h(f(x)) = 2 h(x) + O(1) (the implied constant depends on c).
This gives an a prior bound on the height of all rational points with
finite forward orbits, thus a bound on the number of such points,
and also on the size of their forward orbits.   

References:
  J. Silverman, Rational points on K3 surfaces:  A new canonical height.
Invent. math. 105 (1991), 347-373.

  G. Call and J. Silverman, Canonical heights on varieties with morphisms.
Preprint 1992.