It is tough reading , but it makes sense when considered with the Previous lines & the next lines , even the next Para.
Here is my interpretation :
The Para & next Para ( the Starting line ) is this :
The poor young man hesitated and procrastinated: it cost him such an
effort to broach the subject of terms, to speak of money to a person
who spoke only of feelings and, as it were, of the aristocracy. Yet
he was unwilling to take leave, treating his engagement as settled,
without some more conventional glance in that direction than he could
find an opening for in the manner of the large affable lady who sat
there drawing a pair of soiled gants de Suède through a fat jewelled
hand and, at once pressing and gliding, repeated over and over
everything but the thing he would have liked to hear. He would have
liked to hear the figure of his salary; but just as he was nervously
about to sound that note the little boy came back—the little boy Mrs.
Moreen had sent out of the room to fetch her fan. He came back
without the fan, only with the casual observation that he couldn’t
find it. As he dropped this cynical confession he looked straight and
hard at the candidate for the honour of taking his education in hand.
This personage reflected somewhat grimly that the thing he should have
to teach his little charge would be to appear to address himself to
his mother when he spoke to her—especially not to make her such an
improper answer as that.
When Mrs. Moreen bethought herself of this pretext for getting rid of
their companion Pemberton supposed it was precisely to approach the
delicate subject of his remuneration.
Here we see that Pemberton is the young man who would be hired to tutor a little boy whose mother is Mrs. Moreen. She has sent the little boy out of the room. Pemberton thought that , when the boy was not around , she would discuss Payment Details , but her intention was to talk Positively (& repeatedly !) about the boy who may have some heart ailment.
Till now , she has not yet told him how much he would be paid.
At the Point you refer to , the lady was talking about "higher" (aristocratic !) things & Pemberton wanted to ask about the remuneration. But he was unwilling (or unable) to do that !
He could accept the job & take his leave , without knowing his remuneration. But he was unwilling to do that too !
He wanted to know (or hear) a Pound Amount (or figure) that he would earn.
The lady not talking about such matters , he could not get a chance to initiate or move the conversation in that Direction.
With all that , your Query about this can be attempted :
"...without some more conventional glance in that direction than he could find an opening for in the manner of..."
It is conventional to state what the remuneration will be , in Pounds or other terms.
Here , that was not there. Pemberton wanted to hear some thing , at least "you will get X Pounds a week" or "My secretary will discuss your remuneration" or "we will decide your remuneration at the end of the first week". Here , there was nothing.
Even when Pemberton thought that it would be eventually revealed , the conversation was mostly about "higher" things , never about monetary things.
When Pemberton himself wanted to ask about it , the conversation topics did not let him.
Pemberton did not want to leave without making the conversation move ( or at least glance ) in that Direction : "Oh , about your remuneration : You must talk to my secretary !"
At last , when Pemberton decided to force that topic , he was interrupted by the little boy.