Short answer: Yes in principle, but in practice no one seems to have published a clean, usable value of the “sufficiently large” threshold for the best-known exponents for π; extracting it from the proofs is possible but leads to astronomically large numbers, making it of no practical use. The only widely available completely explicit inequality that holds for all q is still Mahler’s old result (with a very poor exponent).

Details and references:

• Effectivity of the best results

  • Hata’s 1993 paper (μ(π) ≤ 8.0161) and Salikhov’s 2008 paper (μ(π) ≤ 7.6063…) are fully effective: all the auxiliary constants are explicit and, in principle, one can write down an explicit Q0 such that |π − p/q| > 1/q^μ holds for all q ≥ Q0 (for any μ strictly larger than the quoted bound).
  • However, neither paper is written with the goal of presenting a final numerical value of Q0. If you track every estimate tightly enough, you do get a concrete Q0, but it is unimaginably huge (doubly-exponential type growth in intermediate parameters). To the best of my knowledge, no paper or note has published such a Q0 for either Hata’s 8.0161 or Salikhov’s 7.6063… bound.

• A completely explicit bound “for all q”

  • Mahler’s classical argument gives an “unrestricted” inequality of the form |π − p/q| > c/q^42 for all integers p, q ≥ 1, with an explicit positive constant c (some expositions normalize c = 1 after absorbing constants into the exponent; in any case, c is explicit). This is much weaker in exponent, but it is clean and fully explicit.

• A route via π^2

  • There are effective results for π^2 due to Rhin and Viola: μ(π^2) ≤ 5.4412… (mid-1990s). Their constructions are also effective, and one can in principle extract explicit constants. From such a bound one immediately gets an effective bound for π itself by the identity |π − p/q| = |π^2 − (p/q)^2| / (π + p/q), yielding an inequality of the shape |π − p/q| > C/q^{2·5.4412…} for all sufficiently large q, with C and the threshold explicitly computable from the constants in the π^2 bound. The exponent ~10.882… is worse than Hata/Salikhov’s exponents, but this path is another source of completely effective (though still astronomically large) thresholds.

• What to do in practice

  • If you need explicit lower bounds that are meaningful for concrete denominators, continued fractions are far superior: using known partial quotients of π you can certify the best possible lower bounds |π − p/q| for all q up to any practical limit (these bounds are of order 1/(a_{n+1} q^2) near convergents and beat any q^{-μ} with μ > 2 by many orders of magnitude for such q).
  • If you want a globally valid bound for all q, use Mahler’s (or any modern explicit corollary you prefer) and, if desired, splice it with a CF-based verification up to some finite cutoff.

Summary: The modern exponents (8.0161, 7.6063…) do come from effective constructions, so explicit “sufficiently large” thresholds exist in principle, but they are enormous and, as far as I am aware, have not been written down in the literature. For fully explicit statements, Mahler’s old unrestricted inequality remains the go-to reference; for any realistic q one should instead rely on continued fractions. If you really need a concrete threshold from Hata or Salikhov, it can be extracted by a careful bookkeeping of their constants, but expect numbers so large that they won’t be of practical value.