We have no objection to anyone holding a meeting here.

这里不能将现在分词短语 holding a meeting here看作是 anyone的后置定语，而只能分析为anyone holding a meeting here 是动名词复合结构作介词to的宾语，否则意思不通。句意指的是：我们不反对任何人在这里举行会议。而不是：我们不反对任何在这里举行会议的人。

have no objection to 后面可以接 sth，也可以接 doing sth 或 sb / sb’s doing sth，但不接 sb（包括 sb 后接定语的结构）。

have no objection to sth

I have no objection to your proposal. 我对你的提议没有异议。

She had no objection to the plan at first. 一开始她并不反对这一计划。

have no objection to doing sth

He had no objection to getting up early. 他对早起没有反对意见。

I have no objection to hearing your story again. 再听一遍你的故事，我没有异议。

have no objection to sb / sb’s doing sth

I have no objection to her being invited. 我不反对邀请她。

I have no objection to him coming to stay. 我不反对他来小住。

He had no objection to Dinny marrying him. 他不反对丁妮和他结婚。

We had no objection to their getting higher wages. 他们工资高一些我们不反对。