You can think a verb like 考え(て) or して is omitted after と. Volitional + と
is a very common pattern, and it can be translated simply as "thinking ~", "trying to ~" or "in order to ~". Despite its name, quotative-と has a wide variety of usages, and it doesn't have to mark something actually said by someone. Note that none of the examples in this link includes quotative-と that is marking something actually said by someone.
The basic structure of the sentence is this:
日本は、平成16年から定期的に外相会合を開いています。
Japan has held foreign ministerial conferences regularly since Heisei 16.
Everything after 日本は and と is the "quote". The content is Japan's own strategy (i.e., Japan's own thinking) rather than what Japan is thought to be doing or thinking. (There is no passive expression whatsoever in this sentence.)
日本は、「中央アジアへの関与を強めよう」と(考え)、平成16年から定期的に外相会合を開いています。
In order to strengthen the involvement in Middle Asia, Japan has held foreign ministerial conferences regularly since Heisei 16.
"中国やロシアと隣接し、豊富なエネルギー資源がある" is a relative clause that modifies 中央アジア.
日本は、「中国やロシアと隣接し、豊富なエネルギー資源がある中央アジアへの関与を強めよう」と、平成16年から定期的に外相会合を開いています。
In order to strengthen the involvement in Middle Asia, which is adjacent to China and Russia and possesses rich energy resources, Japan has held foreign ministerial conferences regularly since Heisei 16.
So the subjects of 強めよう and (omitted) 考え are both 日本.