But Yorkshire’s other two representatives, Steve Robins (Hull) and Jonathan Thomson (Lindrick) missed out, the latter having to retire because of an injury problem.
Walker added a 75 to his opening 70 to qualify in a tie for 13th place while Brown (74 73) was one of six players who made the last 32 from the seven who tied for 27th place.
Dutch Junior Open champion Brown could have his work cut out in Friday’s match as he faces France’s Romain Langasque, who shot an eight-under-par 64 on Thursday to progress in third place.
Walker faces Ireland’s Conor O’Rourke, both players having qualified with a 145 aggregate.
Robins had begun with a level-par 72, but slipped to a 77 in round two.
Woodsome Hall’s Rochelle Morris went agonisingly close to claiming a place in the match play stages of the Spanish Amateur championship, at Club de Golf Escorpion, in Valencia.
She shot a second-round 76, one more than her opening mark, and it ruled against her in the countback after she had tied 31st in stroke play qualifying.
Her Yorkshire team-mate Olivia Winnings (Rotherham) also opened with a 75, but carded a 78 second time around and also missed out.
Morris stood only one over par for the day following the 14th hole, which she birdied following bogeys at holes three and eight.
However, a double bogey quickly followed at the 15th and she closed the round with another dropped shot.
Winning was quick to negate bogeys at holes two and five with immediate birdies at three and six.
But six further shots got away from her, including three in a row from the 13th, without her being able to claw any back.