He is indignant at suggestions that they were secret agents.

参考译文(1)他对关于他们是特务的暗示很愤慨。

参考译文(2)：有人暗示他们可能是特务，这让他非常气愤。

这句话中 suggest 应该翻译为“暗示”吧？

那后面怎么会是 were 呢？此时不应该用虚拟语气吧？

解答如下：

以上suggestion是“暗示的意思”后面接了一个同位语从句。

He is indignant at suggestions that they were secret agents.

参考译文：有人暗示他们可能是特务，这让他非常气愤。

(1)suggestion 暗示,表明,示意 ,迹象后面使用陈述语气。

We reject any suggestion that the law needs amending.

对于任何暗示该法律需要修改的观点我们都拒绝接受。

There are suggestions that he might be supported by the Socialists. 

有人认为他可能会受到社会党的支持。

There is no suggestion whatsoever that the two sides are any closer to agreeing.

没有任何迹象表明双方将会达成一致。

(2)suggestion 当建议讲"后面接虚拟语气,should +动词原形

His suggestion is what we should leave immediately.

他的建议是我们应该立即离开。

He made the suggestion that the political prisoners (should) be set free.
他建议释放政治犯。

既然suggest是“暗示”，那后面是用了虚拟语气吗？如果是的话，可是suggest做暗示讲不是不能用虚拟语气么？如果不是的话，那用were不就和前面的is时态不一致了么？

继续跟踪回答如下

同位语从句时态不受主句时态限制，只有宾语从句时态必会受到主句时态限制。同位语从句主语是复数they 从句谓语也用复数过去式were形式。从句谓语使用时态是正确的。