Let

  Φ(x)= 1/√(2π)⋅∫_{-∞}^{x} e^{-t²/2},dt ,  x∈ℝ ,

and assume that q₁ , q₂ are rational numbers with

  Φ(q₁)=q₂. (1)

Question: must we have q₁=0 and q₂=1/2 ?

What is known

1. Relation with the error function
     Φ(x)=½(1+erf(x/√2)),  erf(z)= 2/√π ∫₀^{z} e^{-t²},dt .
     Hence (1) is equivalent to

  erf(α)=2q₂−1  with  α = q₁/√2 . (2)

  The point α is algebraic of degree 2 (it is rational times √2) as soon as   q₁ is rational and non–zero. Thus the question becomes

    “Is the value of the error function at a non–zero algebraic argument
     rational?”

Absolutely nothing is currently known that would allow one to prove irrationality, let alone transcendence, of numbers such as
erf(1) ≈ 0.842700…, erf(½) ≈ 0.520500…, etc.

2. Existing transcendence / algebraic-independence theorems
   The powerful results of Gelfond–Schneider, Lindemann–Weierstrass, Siegel-Shidlovskiĭ, … apply to functions that are solutions of certain algebraic differential equations called E–functions. Unfortunately the error function is not an E-function, and these theorems do not tell us anything about its values at algebraic points. At present there is no general theorem that would even prove that erf(1) is irrational.

3. What is actually proven
   The only algebraic point at which a rational value of Φ (or erf) is known is the obvious one:

   Φ(0)=½  ⇔ erf(0)=0.
   

   No other example with both the argument and the value algebraic is known, and no proof excluding further examples is available.

Consequence for the question

Because no irrationality (let alone transcendence) result is known for Φ(q) at a non–zero rational q, the implication (1) ⇒ (q₁=0, q₂=½) is an open problem. It is widely believed that Φ(q) is transcendental for every non-zero algebraic q—which would, in particular, make it impossible for (1) to hold with q₁≠0—but this has not been proved.

Therefore, at the current state of knowledge:

We do not know whether there exist non-zero rational numbers q₁ for which Φ(q₁) is rational. The only rational value that is proved to occur is Φ(0)=1/2.

In other words, the question you ask is unsolved; the implication you proposed is expected to be true, but no proof is known.