British number one Konta suffered a straight-sets defeat by Zheng Saisai.
Konta, ranked 18 in the world, lost 6-4 7-5 to the Chinese, who is 44 places below her in the world rankings.
Zheng will now face Hong Kong-born Moore, 23, who beat American world number 67 Christina McHale 6-2 6-2 to reach the quarter-finals.
Konta needed a medical time out three games into the second set for a "niggle" on her pelvis.
But following the match, the 25-year-old said she expected to play in the Wimbledon warm-up events in Birmingham and Eastbourne.
The 25-year-old was below the form which helped her become the first British woman since 1983 to break into the top 20, making four double faults and winning just 38% of points on her second serve.
British number two Heather Watson and compatriot Laura Robson were knocked out in the first round.
Johanna Konta exited the Nottingham Open in the second round to leave qualifier Tara Moore as the only British player in the competition.