To determine the mass number and atomic number of the final product $Q$ in the provided nuclear reactions, we need to analyze the impact of each type of decay (alpha, positron emission, beta-minus decay, and electron capture) on the mass number and atomic number of the initial element $X = { }_{82}^{290} X$:

1. Alpha decay ($\alpha$ decay): In alpha decay, an alpha particle (which is a $^4_2He$ nucleus) is emitted. This reduces the mass number by 4 units and the atomic number by 2 units.

$$ { }_{82}^{290} X \xrightarrow{\alpha} { }_{80}^{286} Y $$

2. Beta plus decay (positron emission, $e^{+}$): During positron emission, a proton in the nucleus is transformed into a neutron, and a positron is emitted. This decreases the atomic number by 1 but does not change the mass number.

$$ { }_{80}^{286} Y \xrightarrow{e^{+}} { }_{79}^{286} Z $$

3. Beta-minus decay ($\beta^{-}$ decay): In a beta-minus decay, a neutron in the nucleus converts into a proton and an electron (beta particle) and an antineutrino are emitted. This results in an increase in the atomic number by 1, while the mass number remains unchanged.

$$ { }_{79}^{286} Z \xrightarrow{\beta^{-}} { }_{80}^{286} P $$

4. Electron capture ($e^{-}$ capture): During electron capture, an atomic electron is absorbed by the nucleus, causing a proton to convert into a neutron. This process decreases the atomic number by 1 without altering the mass number.

$$ { }_{80}^{286} P \xrightarrow{e^{-}} { }_{79}^{286} Q $$

From the calculations above, the mass number of $Q$ is 286, and its atomic number is 79. Comparing these values with the multiple choices:

Option A: Mass number = 280, Atomic number = 81
Option B: Mass number = 286, Atomic number = 80
Option C: Mass number = 288, Atomic number = 82
Option D: Mass number = 286, Atomic number = 81

None of these match exactly, implying an error in the problem or the answer options. Based on the correct atomic number calculations and assuming option D is meant to be atomic number 79, then the corrected choice would have been:

D: Mass number = 286, Atomic number = 79