To start, it is important to notice that a statement like "For each x antiquarks, there are x+1 quarks" is vacuous unless you specify the instant at which you are looking at the system. To give an example, if you start with 100 antiquarks for every 101 quark, there will be a time when you'll have 50 for every 51, and finally 0 for every 1.
This said, I'd recommend to disregard this unfortunate statement in an otherwise awesome review and turn to Kolb and Turner's The Early Universe, where a similar comment is made (near equation 6.2, page 159):
Although the baryon asymmetry is maximal today,
i.e., no antimatter, at high temperatures ($T \gtrsim 1\,\textrm{GeV}$) thermal quark-antiquark pairs were present in great numbers ($n_q \sim n_{\bar q} \sim n_\gamma$), so that the
baryon asymmetry observed today corresponds to a tiny quark-antiquark
asymmetry at early times ($t \lesssim 10^{-6}\,\textrm{s}$):
$$\frac{n_q-n_{\bar q}}{n_q} \simeq 3 \times 10^{-8}\,.$$
That is, for every 30 million antiquarks, there were 30 million and 1 quarks present! A very tiny asymmetry indeed.
Unlike Davidson, Nardi and Nir, these authors refer to a specific time, namely around and before $t \sim 10^{-6}$ seconds, corresponding to temperatures about and above $1\,\textrm{GeV}$.
Lets take a stab at computing their quoted value. At such time in the history of the Universe, particles such as the light quarks move relativistically, and in this limit $n_q \simeq n_{\bar q} \simeq \frac{3}{4} n_\gamma$ (see eq. 3.52 of the same book).
Additionally, the density of photons $n_\gamma$ and the entropy density $s$ are related at all times by (chapter 3.4):
$$s = \frac{\pi^4\, g_{*s}}{45\, \zeta(3)} \,n_\gamma\simeq 1.8\, g_{*s}\,n_\gamma\,.$$
Here $g_{*s}$ is a function of time. For temperatures around $1\,\textrm{GeV}$ one has $g_{*s} \simeq 62$ according to this detailed StackExchange answer. Hence,
$$
\frac{n_q - n_{\bar q}}{n_q}
\simeq \frac{3 n_B - 3 n_{\bar B}}{\frac{3}{4} n_\gamma}
\simeq 4 \frac{n_B - n_{\bar B}}{n_\gamma}
\simeq 4 \frac{s}{n_\gamma} \frac{n_B - n_{\bar B}}{s}
\simeq 4 \, (1.8 \cdot 62) \, Y_{\Delta B}\,,$$
where $Y_{\Delta B} \equiv (n_B - n_{\bar B})/{s}$ is measured to be $\simeq 9 \times 10^{-11}$ today according to (1.2) of Davidson, Nardi and Nir. The nice thing is that this quantity is unchanged from the moment baryon violating interactions become inactive (way before $t \sim 10^{-6}\,\textrm{s}$ in a log scale), and assuming no entropy production.
So we can plug it in and get:
$$
\frac{n_q - n_{\bar q}}{n_q}
\simeq 4 \cdot 1.8 \cdot 62 \cdot 9 \times 10^{-11} \simeq 4 \times 10^{-8}
\,,$$
which is in good enough agreement with what is quoted in the book :)
