Klugman went an early break down against Bouzkova but responded well, showing some nifty defence before sending a passing forehand winner past the Czech to get back on serve.
She also created three break points for a 5-3 lead but Bouzkova saved them all, before winning the final three games to close out the set.
Klugman became more error-prone in the second set, finishing with 45 unforced errors to 29 winners and committing nine double faults as Bouzkova took control.
However, it has still been an important week for the highly-rated Briton as she navigates the move from the junior to senior ranks.
“She’s such a quality player and there are glimpses of some good stuff in there from me, so there are so many positives to take,” Klugman said of Bouzkova.
“Mentally, I was great today and the crowd was amazing playing at home.”
At Queen’s, fourth seed Alejandro Davidovich Fokina beat France’s Corentin Moutet to reach the quarter-finals.
The Spaniard has a superstition of wearing one black sock and one white sock – and joked he may ask for a dispensation from Wimbledon’s all-white dress code.
“It was a long time ago, I wore odd socks and won that match and now I keep wearing them,” he said.
“I will have to ask Wimbledon if I can keep wearing odd socks.”
Davidovich Fokina will face 2024 champion Tommy Paul for a place in the semi-finals.
Elsewhere, French Open champion Alexander Zverev reached the quarter-finals at Halle with a 6-3 7-6 (7-4) win over fellow German Yannick Hanfmann.
Big-serving American Ben Shelton also advanced to set up a meeting with compatriot Taylor Fritz, who he beat in the Stuttgart Open final on Sunday.
