QT-blocks involve the other two fabfour series that are always part of the set of series of a fractional approach, the quartus and the tertius.
We've defined a fraction's value 'v' = (numerator)²-n*(denominator)². With regard to this, qt-blocks come in three kinds:

1. A multiple of the tertius in the numerator, a multiple of the quartus in the denominator.
   In this case v = t²-nq² > 1 and the value of the fraction is above the root.
2. A multiple of the quartus in the numerator, a multiple of the tertius in the denominator.
   In this case v = q²-nt² < 0 and the value of the fraction is below the root.
3. Both of the above in successive fractions.
   In this case the t/q fraction precedes the q/t fraction and v₁*v₂ = -n.

From the first qt-block ⇒ the first non trivial sp-block
The following formulas hold for both carréphylic and carréphobic numbers. For the former they are inconsequential because the diophantine equation s²-np² = 1 can be solved. For the latter they allow the calculation of 's' and 'p' in cases where a direct search fails.
As far as this investigation is concerned, I used them to determine the sp-blocks of √61 (1766319049 / 226153980) and √109 (158070671986249 / 15140424455100), both of which were out of range of Ed's fractional approximations program.

Conjecture

• A fraction is part of a (semi-)symmetric series if and only if v | 2*(numerator).
  If v = 1 we have the first fraction of a sp-block, otherwise it is a qt-block.

Let's take a t/q fraction, so v > 1 and v | 2t. Since t²-nq²= v, the distance of nq² to the nearest square (t²) is v, which divides 2t. Hence nq² is carréphylic, so for the sp-series linked to the approach of its root: s = (2t²-v)/v and p = 2t/v.
Since √nq² = Q√n, s/p is an approximation of Q√n, so for √n itself:

• s = (2t²-v)/v and p = 2qt/v

A similar reasoning applies to q/t fractions:

• s = -(2q²-v)/v and p = -2tq/v