Abstract : The SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS), a five-year spectroscopic survey of 10,000 deg2, achieved first light in late 2009. One of the key goals of BOSS is to measure the signature of baryon acoustic oscillations (BAOs) in the distribution of Lyα absorption from the spectra of a sample of ~150,000 z > 2.2 quasars. Along with measuring the angular diameter distance at z ≈ 2.5, BOSS will provide the first direct measurement of the expansion rate of the universe at z > 2. One of the biggest challenges in achieving this goal is an efficient target selection algorithm for quasars in the redshift range 2.2 < z < 3.5, where their colors tend to overlap those of the far more numerous stars. During the first year of the BOSS survey, quasar target selection (QTS) methods were developed and tested to meet the requirement of delivering at least 15 quasars deg-2 in this redshift range, with a goal of 20 out of 40 targets deg-2 allocated to the quasar survey. To achieve these surface densities, the magnitude limit of the quasar targets was set at g <= 22.0 or r <= 21.85. While detection of the BAO signature in the distribution of Lyα absorption in quasar spectra does not require a uniform target selection algorithm, many other astrophysical studies do. We have therefore defined a uniformly selected subsample of 20 targets deg-2, for which the selection efficiency is just over 50% (~10 z > 2.20 quasars deg-2). This "CORE" subsample will be fixed for Years Two through Five of the survey. For the remaining 20 targets deg-2, we will continue to develop improved selection techniques, including the use of additional data sets beyond the Sloan Digital Sky Survey (SDSS) imaging data. In this paper, we describe the evolution and implementation of the BOSS QTS algorithms during the first two years of BOSS operations (through 2011 July), in support of the science investigations based on these data, and we analyze the spectra obtained during the first year. During this year, 11,263 new z > 2.20 quasars were spectroscopically confirmed by BOSS, roughly double the number of previously known quasars with z > 2.20. Our current algorithms select an average of 15 z > 2.20 quasars deg-2 from 40 targets deg-2 using single-epoch SDSS imaging. Multi-epoch optical data and data at other wavelengths can further improve the efficiency and completeness of BOSS QTS.